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Gaussian Elimination. Chemical Engineering Majors Author(s): Autar Kaw http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates. Naïve Gauss Elimination http://numericalmethods.eng.usf.edu. Naïve Gaussian Elimination.

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Gaussian Elimination

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Gaussian Elimination

Chemical Engineering Majors

Author(s): Autar Kaw

http://numericalmethods.eng.usf.edu

Transforming Numerical Methods Education for STEM Undergraduates


Naïve Gauss Eliminationhttp://numericalmethods.eng.usf.edu


Naïve Gaussian Elimination

A method to solve simultaneous linear equations of the form [A][X]=[C]

Two steps

1. Forward Elimination

2. Back Substitution


Forward Elimination

The goal of forward elimination is to transform the coefficient matrix into an upper triangular matrix


Forward Elimination

A set of n equations and n unknowns

. .

. .

. .

(n-1) steps of forward elimination


Forward Elimination

Step 1

For Equation 2, divide Equation 1 by and multiply by .


Forward Elimination

Subtract the result from Equation 2.

_________________________________________________

or


Forward Elimination

Repeat this procedure for the remaining equations to reduce the set of equations as

. ..

. ..

. ..

End of Step 1


Forward Elimination

Step 2

Repeat the same procedure for the 3rd term of Equation 3.

. .

. .

. .

End of Step 2


Forward Elimination

At the end of (n-1) Forward Elimination steps, the system of equations will look like

. .

. .

. .

End of Step (n-1)


Matrix Form at End of Forward Elimination


Back Substitution

Solve each equation starting from the last equation

Example of a system of 3 equations


Back Substitution Starting Eqns

. .

. .

. .


Back Substitution

Start with the last equation because it has only one unknown


Back Substitution


THE END

http://numericalmethods.eng.usf.edu


Naïve Gauss EliminationExamplehttp://numericalmethods.eng.usf.edu


Example: Liquid-Liquid Extraction

A liquid-liquid extraction process conducted in the Electrochemical Materials Laboratory involved the extraction of nickel from the aqueous phase into an organic phase. A typical set of experimental data from the laboratory is given below:

Assuming g is the amount of Ni in organic phase and a is the amount of Ni in the aqueous phase, the quadratic interpolant that estimates g is given by


Example: Liquid-Liquid Extraction

The solution for the unknowns x1, x2, and x3 is given by

Find the values of x1, x2, and x3 using Naïve Gauss Elimination. Estimate the amount of nickel in organic phase when 2.3 g/l is in the aqueous phase using quadratic interpolation.


Number of Steps of Forward Elimination

Number of steps of forward elimination is

(n-1)=(3-1)=2


Example: Liquid-Liquid Extraction

Solution

Forward Elimination: Step 1

Yields


Example: Liquid-Liquid Extraction

Forward Elimination: Step 1

Yields


Example: Liquid-Liquid Extraction

Forward Elimination: Step 2

Yields

This is now ready for Back Substitution


Example: Liquid-Liquid Extraction

Back Substitution: Solve for x3 using the third equation


Example: Liquid-Liquid Extraction

Back Substitution: Solve for x2 using the second equation


Example: Liquid-Liquid Extraction

Back Substitution:Solve for x1 using the first equation


Example: Liquid-Liquid Extraction

The solution vector is

The polynomial that passes through the three data points is then

Where g is the amount of nickel in the organic phase and a is the amount of in the aqueous phase.


Example: Liquid-Liquid Extraction

When 2.3 g/l is in the aqueous phase, using quadratic interpolation, the estimated amount of nickel in the organic phase is


THE END

http://numericalmethods.eng.usf.edu


Naïve Gauss EliminationPitfallshttp://numericalmethods.eng.usf.edu


Pitfall#1. Division by zero


Is division by zero an issue here?


Is division by zero an issue here? YES

Division by zero is a possibility at any step of forward elimination


Pitfall#2. Large Round-off Errors

Exact Solution


Pitfall#2. Large Round-off Errors

Solve it on a computer using 6 significant digits with chopping


Pitfall#2. Large Round-off Errors

Solve it on a computer using 5 significant digits with chopping

Is there a way to reduce the round off error?


Avoiding Pitfalls

  • Increase the number of significant digits

  • Decreases round-off error

  • Does not avoid division by zero


Avoiding Pitfalls

  • Gaussian Elimination with Partial Pivoting

  • Avoids division by zero

  • Reduces round off error


THE END

http://numericalmethods.eng.usf.edu


Gauss Elimination with Partial Pivotinghttp://numericalmethods.eng.usf.edu


Pitfalls of Naïve Gauss Elimination

  • Possible division by zero

  • Large round-off errors


Avoiding Pitfalls

  • Increase the number of significant digits

  • Decreases round-off error

  • Does not avoid division by zero


Avoiding Pitfalls

  • Gaussian Elimination with Partial Pivoting

  • Avoids division by zero

  • Reduces round off error


What is Different About Partial Pivoting?

At the beginning of the kth step of forward elimination, find the maximum of

If the maximum of the values is

in the p th row,

then switch rows p and k.


Matrix Form at Beginning of 2nd Step of Forward Elimination


Example (2nd step of FE)

Which two rows would you switch?


Example (2nd step of FE)

Switched Rows


Gaussian Elimination with Partial Pivoting

A method to solve simultaneous linear equations of the form [A][X]=[C]

Two steps

1. Forward Elimination

2. Back Substitution


Forward Elimination

Same as naïve Gauss elimination method except that we switch rows before each of the (n-1) steps of forward elimination.


Example: Matrix Form at Beginning of 2nd Step of Forward Elimination


Matrix Form at End of Forward Elimination


Back Substitution Starting Eqns

. .

. .

. .


Back Substitution


THE END

http://numericalmethods.eng.usf.edu


Gauss Elimination with Partial PivotingExamplehttp://numericalmethods.eng.usf.edu


Example 2

Solve the following set of equations by Gaussian elimination with partial pivoting


Example 2 Cont.

  • Forward Elimination

  • Back Substitution


Forward Elimination


Number of Steps of Forward Elimination

Number of steps of forward elimination is (n-1)=(3-1)=2


Forward Elimination: Step 1

Examine absolute values of first column, first row

and below.

  • Largest absolute value is 144 and exists in row 3.

  • Switch row 1 and row 3.


Forward Elimination: Step 1 (cont.)

Divide Equation 1 by 144 and

multiply it by 64, .

.

Subtract the result from Equation 2

Substitute new equation for Equation 2


Forward Elimination: Step 1 (cont.)

Divide Equation 1 by 144 and

multiply it by 25, .

.

Subtract the result from Equation 3

Substitute new equation for Equation 3


Forward Elimination: Step 2

Examine absolute values of second column, second row

and below.

  • Largest absolute value is 2.917 and exists in row 3.

  • Switch row 2 and row 3.


Forward Elimination: Step 2 (cont.)

Divide Equation 2 by 2.917 and

multiply it by 2.667,

.

Subtract the result from Equation 3

Substitute new equation for Equation 3


Back Substitution


Back Substitution

Solving for a3


Back Substitution (cont.)

Solving for a2


Back Substitution (cont.)

Solving for a1


Gaussian Elimination with Partial Pivoting Solution


Gauss Elimination with Partial PivotingAnother Examplehttp://numericalmethods.eng.usf.edu


Partial Pivoting: Example

Consider the system of equations

In matrix form

=

Solve using Gaussian Elimination with Partial Pivoting using five significant digits with chopping


Partial Pivoting: Example

Forward Elimination: Step 1

Examining the values of the first column

|10|, |-3|, and |5| or 10, 3, and 5

The largest absolute value is 10, which means, to follow the rules of Partial Pivoting, we switch row1 with row1.

Performing Forward Elimination


Partial Pivoting: Example

Forward Elimination: Step 2

Examining the values of the first column

|-0.001| and |2.5| or 0.0001 and 2.5

The largest absolute value is 2.5, so row 2 is switched with row 3

Performing the row swap


Partial Pivoting: Example

Forward Elimination: Step 2

Performing the Forward Elimination results in:


Partial Pivoting: Example

Back Substitution

Solving the equations through back substitution


Partial Pivoting: Example

Compare the calculated and exact solution

The fact that they are equal is coincidence, but it does illustrate the advantage of Partial Pivoting


THE END

http://numericalmethods.eng.usf.edu


Determinant of a Square MatrixUsing Naïve Gauss EliminationExamplehttp://numericalmethods.eng.usf.edu


Theorem of Determinants

If a multiple of one row of [A]nxn is added or subtracted to another row of [A]nxn to result in [B]nxn then det(A)=det(B)


Theorem of Determinants

The determinant of an upper triangular matrix [A]nxn is given by


Forward Elimination of a Square Matrix

Using forward elimination to transform [A]nxn to an upper triangular matrix, [U]nxn.


Example

Using naïve Gaussian elimination find the determinant of the following square matrix.


Forward Elimination


Forward Elimination: Step 1

Divide Equation 1 by 25 and

multiply it by 64, .

.

Subtract the result from Equation 2

Substitute new equation for Equation 2


Forward Elimination: Step 1 (cont.)

Divide Equation 1 by 25 and

multiply it by 144, .

.

Subtract the result from Equation 3

Substitute new equation for Equation 3


Forward Elimination: Step 2

Divide Equation 2 by −4.8

and multiply it by −16.8,

.

.

Subtract the result from Equation 3

Substitute new equation for Equation 3


Finding the Determinant

After forward elimination

.


Summary

  • Forward Elimination

  • Back Substitution

  • Pitfalls

  • Improvements

  • Partial Pivoting

  • Determinant of a Matrix


Additional Resources

For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit

http://numericalmethods.eng.usf.edu/topics/gaussian_elimination.html


THE END

http://numericalmethods.eng.usf.edu


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